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Matching statistic: St000647
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(load all 3 compositions to match this statistic)
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 2 = 3 - 1
[4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 1 = 2 - 1
[1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 1 = 2 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 1 = 2 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => 2 = 3 - 1
[1,4,3,2,5] => 0 = 1 - 1
[1,4,3,5,2] => 1 = 2 - 1
[1,4,5,2,3] => 1 = 2 - 1
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St000325
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 3
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => 2
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 3
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => 2
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000021
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => 2 = 3 - 1
[4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 1 = 2 - 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000155
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 2 = 3 - 1
[3,2,1,4] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 2 = 3 - 1
[3,4,1,2] => [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 1 = 2 - 1
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000662
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => 2 = 3 - 1
[4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 1 = 2 - 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001683
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => 2 = 3 - 1
[2,1,4,3] => [1,4,2,3] => 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => [1,3,4,2,5] => 2 = 3 - 1
[1,3,4,5,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 2 = 3 - 1
[1,3,5,4,2] => [1,3,5,2,4] => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => 2 = 3 - 1
[1,4,3,2,5] => [1,4,2,5,3] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,2,3,5] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 2 = 3 - 1
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St000010
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [2] => [2]
=> 1
[2,1] => [2] => [2]
=> 1
[1,2,3] => [3] => [3]
=> 1
[1,3,2] => [1,2] => [2,1]
=> 2
[2,1,3] => [3] => [3]
=> 1
[2,3,1] => [3] => [3]
=> 1
[3,1,2] => [3] => [3]
=> 1
[3,2,1] => [2,1] => [2,1]
=> 2
[1,2,3,4] => [4] => [4]
=> 1
[1,2,4,3] => [2,2] => [2,2]
=> 2
[1,3,2,4] => [1,3] => [3,1]
=> 2
[1,3,4,2] => [1,3] => [3,1]
=> 2
[1,4,2,3] => [1,3] => [3,1]
=> 2
[1,4,3,2] => [1,2,1] => [2,1,1]
=> 3
[2,1,3,4] => [4] => [4]
=> 1
[2,1,4,3] => [2,2] => [2,2]
=> 2
[2,3,1,4] => [4] => [4]
=> 1
[2,3,4,1] => [4] => [4]
=> 1
[2,4,1,3] => [4] => [4]
=> 1
[2,4,3,1] => [3,1] => [3,1]
=> 2
[3,1,2,4] => [4] => [4]
=> 1
[3,1,4,2] => [2,2] => [2,2]
=> 2
[3,2,1,4] => [2,2] => [2,2]
=> 2
[3,2,4,1] => [2,2] => [2,2]
=> 2
[3,4,1,2] => [4] => [4]
=> 1
[3,4,2,1] => [3,1] => [3,1]
=> 2
[4,1,2,3] => [4] => [4]
=> 1
[4,1,3,2] => [3,1] => [3,1]
=> 2
[4,2,1,3] => [2,2] => [2,2]
=> 2
[4,2,3,1] => [3,1] => [3,1]
=> 2
[4,3,1,2] => [1,3] => [3,1]
=> 2
[4,3,2,1] => [1,2,1] => [2,1,1]
=> 3
[1,2,3,4,5] => [5] => [5]
=> 1
[1,2,3,5,4] => [3,2] => [3,2]
=> 2
[1,2,4,3,5] => [2,3] => [3,2]
=> 2
[1,2,4,5,3] => [2,3] => [3,2]
=> 2
[1,2,5,3,4] => [2,3] => [3,2]
=> 2
[1,2,5,4,3] => [2,2,1] => [2,2,1]
=> 3
[1,3,2,4,5] => [1,4] => [4,1]
=> 2
[1,3,2,5,4] => [1,2,2] => [2,2,1]
=> 3
[1,3,4,2,5] => [1,4] => [4,1]
=> 2
[1,3,4,5,2] => [1,4] => [4,1]
=> 2
[1,3,5,2,4] => [1,4] => [4,1]
=> 2
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> 3
[1,4,2,3,5] => [1,4] => [4,1]
=> 2
[1,4,2,5,3] => [1,2,2] => [2,2,1]
=> 3
[1,4,3,2,5] => [1,2,2] => [2,2,1]
=> 3
[1,4,3,5,2] => [1,2,2] => [2,2,1]
=> 3
[1,4,5,2,3] => [1,4] => [4,1]
=> 2
Description
The length of the partition.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000015
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The number of peaks of a Dyck path.
The following 308 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001427The number of descents of a signed permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000204The number of internal nodes of a binary tree. St000211The rank of the set partition. St000223The number of nestings in the permutation. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000356The number of occurrences of the pattern 13-2. St000362The size of a minimal vertex cover of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000536The pathwidth of a graph. St000703The number of deficiencies of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000025The number of initial rises of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000167The number of leaves of an ordered tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000228The size of a partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000286The number of connected components of the complement of a graph. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000005The bounce statistic of a Dyck path. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000141The maximum drop size of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000246The number of non-inversions of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000312The number of leaves in a graph. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000358The number of occurrences of the pattern 31-2. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000632The jump number of the poset. St000672The number of minimal elements in Bruhat order not less than the permutation. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001357The maximal degree of a regular spanning subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001729The number of visible descents of a permutation. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St000652The maximal difference between successive positions of a permutation. St000702The number of weak deficiencies of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000691The number of changes of a binary word. St000829The Ulam distance of a permutation to the identity permutation. St000216The absolute length of a permutation. St000668The least common multiple of the parts of the partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000708The product of the parts of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St001128The exponens consonantiae of a partition. St001246The maximal difference between two consecutive entries of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000809The reduced reflection length of the permutation. St000836The number of descents of distance 2 of a permutation. St000984The number of boxes below precisely one peak. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001964The interval resolution global dimension of a poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000259The diameter of a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001624The breadth of a lattice. St000937The number of positive values of the symmetric group character corresponding to the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001435The number of missing boxes in the first row. St001863The number of weak excedances of a signed permutation. St001862The number of crossings of a signed permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000640The rank of the largest boolean interval in a poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001820The size of the image of the pop stack sorting operator. St000264The girth of a graph, which is not a tree. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001822The number of alignments of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001875The number of simple modules with projective dimension at most 1. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001095The number of non-isomorphic posets with precisely one further covering relation. St000916The packing number of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001272The number of graphs with the same degree sequence. St001393The induced matching number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path.
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